Apparatus and methods for quantum computing and machine learning

ABSTRACT

An apparatus includes a plurality of processing layers coupled in series. Each processing layer in the plurality of processing layers includes a Gaussian unit configured to perform a linear transformation on an input signal including a plurality of optical modes. The Gaussian unit includes a network of interconnected beamsplitters and phase shifters and a plurality of squeezers operatively coupled to the network of interconnected beamsplitters and phase shifters. Each processing layer also includes a plurality of nonlinear gates operatively coupled to the Gaussian unit and configured to perform a nonlinear transformation on the plurality of optical modes. The apparatus also includes a controller operatively coupled to the plurality of processing layers and configured to control a setting of the plurality of processing layers.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of U.S. patent application Ser. No.16/444,624, filed Jun. 18, 2019 and titled “Apparatus and Methods forQuantum Computing and Machine Learning,” the disclosure of which isincorporated herein by reference in its entirety.

FIELD

One or more embodiments relate to photonic circuits for quantumcomputing and machine learning.

BACKGROUND

Quantum computers, which rely on quantum effects, such as superposition,interference, and entanglement, to perform computation, are a promisingtool to implement certain emerging applications, including data fitting,principal component analysis, Bayesian inference, Monte Carlo methods,support vector machines, Boltzmann machines, and recommendation systems.On the classical computing side, the development of classical neuralnetworks (CNNs), such as deep learning, also shows great potential dueto new software libraries and powerful special-purpose computationalhardware. Instead of bit registers in digital computing, the fundamentalcomputational units in deep learning are continuous vectors and tensorsthat are transformed in high dimensional spaces. At the moment, thesecontinuous computations are usually approximated using conventionaldigital computers. Quantum computers may provide an intriguing platformfor exploring new types of neural networks, such as quantum neuralnetworks (QNNs) or hybrid classical-quantum neural networks. To date,however, no known hardware platform is available to implement universalquantum computation and artificial neural networks (both classical andquantum) at the same time.

SUMMARY

Some embodiments described herein relate generally to photonic circuitsfor quantum computing and machine learning, and, in particular, to auniversal platform that is capable of implementing classical neuralnetworks (CNNs), quantum computing (QC), and quantum neural networks(QNNs). In some embodiments, an apparatus includes a plurality ofprocessing layers coupled in series. Each processing layer in theplurality of processing layers includes a Gaussian unit configured toperform a linear transformation on an input signal including a pluralityof optical modes. The Gaussian unit includes a network of interconnectedbeamsplitters and phase shifters and a plurality of squeezersoperatively coupled to the network of interconnected beamsplitters andphase shifters. Each processing layer also includes a plurality ofnonlinear gates operatively coupled to the Gaussian unit and configuredto perform a nonlinear transformation on the plurality of optical modes.The apparatus also includes a controller operatively coupled to theplurality of processing layers and configured to control a setting ofthe plurality of processing layers.

In some embodiments, a method includes propagating an input signalthrough a plurality of processing layers connected in series. The inputsignal includes a plurality of optical modes. The propagation of theinput signal through the plurality of processing layers includesperforming a linear transformation on the plurality of optical modesusing a Gaussian unit and performing a nonlinear transformation on theplurality of optical modes using a plurality of nonlinear gates. TheGaussian unit includes a network of interconnected beamsplitters andphase shifters and a plurality of squeezers that is operatively coupledto the network of interconnected beamsplitters and phase shifters. Themethod also includes sending an output signal from the plurality ofprocessing layers.

In some embodiments, a reconfigurable computing device includes aplurality of processing layers coupled in series and configured toreceive an input signal including a plurality of optical modes. Eachprocessing layer includes a Gaussian unit configured to perform a lineartransformation on the plurality of optical modes and a plurality ofnonlinear gates configured to perform a nonlinear transformation overthe plurality of optical modes. The Gaussian unit includes a network ofinterconnected beamsplitters and phase shifters, a plurality ofsqueezers operatively coupled to the network of interconnectedbeamsplitters and phase shifters, and a plurality of displacersoperatively coupled to the plurality of squeezers and the network ofinterconnected beamsplitters and phase shifters. The reconfigurablecomputing device also includes a controller operatively coupled to theplurality of processing layers and configured to switch the apparatusbetween a first mode to implement a classical neural network, a secondmode to implement a quantum computation, and a third mode to implement aquantum neural network. During the first mode, a first portion of thenetwork of interconnected beamsplitters and phase shifters is configuredto form a first interferometer to apply a first phase-lesstransformation on the plurality of optical modes, the plurality ofsqueezers is configured to apply a position-only squeezing to theplurality of optical modes, a second portion of the network ofinterconnected beamsplitters and phase shifters is configured to form asecond interferometer to apply a second phase-less transformation on theplurality of optical modes, the plurality of displacers is configured toapply a position-only displacement to the plurality of optical modes,and the plurality of nonlinear gates is configured to apply thenonlinear transformation between a first set of position eigenstates anda second set of position eigenstates of the plurality of optical modes.During the second mode, at least one of the Gaussian unit or theplurality of nonlinear gates is configured to create an entanglement ora superposition in the plurality of optical modes. During the thirdmode, at least one of the Gaussian unit or the plurality of nonlineargates is configured to create the entanglement or the superposition inthe plurality of optical modes, and the controller is configured tochange a setting of the plurality of processing layers based on anoutput signal from the plurality of processing layers.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawings primarily are for illustration purposes and are notintended to limit the scope of the subject matter described herein. Thedrawings are not necessarily to scale; in some instances, variousaspects of the disclosed subject matter disclosed herein may be shownexaggerated or enlarged in the drawings to facilitate an understandingof different features. In the drawings, like reference charactersgenerally refer to like features (e.g., functionally similar and/orstructurally similar elements).

FIG. 1 shows a schematic of an apparatus for quantum computing andmachine learning, according to an embodiment.

FIG. 2 shows a schematic of an apparatus including two interferometersat each processing layer for quantum computing and machine learning,according to an embodiment.

FIG. 3 shows a schematic of an apparatus including a feedback loop forquantum computing and machine learning, according to an embodiment.

FIG. 4 is a flowchart illustrating a method of quantum computing with afeedback loop, according to an embodiment.

FIGS. 5A-5D shows schematics of systems for quantum-classical hybridcomputing, according to embodiments.

FIG. 6 shows a schematic of a classical neural network (CNN) implementedby apparatus shown in FIGS. 1-3 , according to an embodiment.

FIG. 7 shows a schematic of a quantum neural network (QNN) implementedby apparatus shown in FIGS. 1-3 , according to an embodiment.

FIG. 8 is a flowchart illustrating a method of evaluating gradients withrespect to a QNN, according to an embodiment.

FIG. 9 shows a schematic of a convolutional processing layer that can beused for quantum computing and machine learning, according to anembodiment.

FIG. 10 shows a recurrent processing layer that can be used for quantumcomputing and machine learning, according to an embodiment.

FIG. 11 shows a residual processing layer that can be used for quantumcomputing and machine learning, according to an embodiment.

FIG. 12 is a flowchart illustrating a method of quantum computing andmachine learning, according to an embodiment.

DETAILED DESCRIPTION

Some embodiments described herein relate to photonic circuits that arecapable of implementing universal quantum computation and artificialneural networks, both classical and quantum. The circuits areconstructed by photonic quantum gates, such as squeezing gates (alsoreferred to as squeezers), displacement gates (also referred to asdisplacers), beamsplitters, phase gates (also referred to as phaseshifters), and nonlinear-optical gates. These photonic quantum gates arecombined into a programmable layer that is configured to process lightthat passes through the layer (e.g., linear and/or nonlineartransformations) and thereby implement quantum computing (QC), classicalneural network (CNN), and/or quantum neural network (QNN).

Quantum computers function in a fundamentally different way compared toclassical computing devices. Using quantum mechanics, quantum computersshow great potential for increased computational power by, e.g.,reducing processing time and handling bigger data. The manufacture of aquantum computer typically involves the ability to implement anddynamically control quantum gates, which can be seen as the analog ofclassical electronic components, such as transistors and resistors. Forlarge scale quantum computers, it can also be beneficial to implementmicro-scale and/or nano-scale quantum gates (e.g., on integratedcircuits), as well as being able to protect these gates from noise.

Photons (i.e., particles of light) can be a promising medium to probeand exploit quantum mechanics. This approach has a number of advantages,including low noise, capability of room temperature operation, fast andenergy-efficient computation, and scalability using developed photonicquantum gates (e.g., squeezers, displacers, beamsplitters, phaseshifters, and nonlinear gates). These gates represent the transformationthey carry out on quantum states, and they can be implemented physicallyin a variety of ways, at both the macroscopic scale and the micro- andnano-scale. These photonic quantum gates can be manufactured usingintegrated photonics circuits and/or bulk optical elements.

Artificial neural networks are an alternative paradigm to conventionalcomputing, enacting algorithms by passing information through a seriesof programmable layers. The information at each layer is typicallyrepresented by a continuous valued vector, and each consecutive layerapplies a linear affine transformation followed by a non-linearactivation function. Neural networks are capable of universal functionapproximation, and are particularly useful as flexible functionapproximators in machine learning and optimization. As used herein, CNNsrefer to artificial neural networks based on classical computing (e.g.,using classical effects of photons), and QNNs refer to artificial neuralnetwork based on quantum computing (e.g., using quantum effects ofphotons, such as entanglement and superposition).

In general, the overall goal of a QNN is to duplicate, either exactly orwith very high fidelity, the transformations or properties of artificialneural networks using quantum computers. This means to achieve thetransformation x→φ(Wx+b) directly on a quantum computer, where x isinput states, W is a weight matrix, b is a bias vector, and φ is anonlinear function. In other words, a QNN is to implement the powerfulnonlinear and nonreversible functions provided by artificial neuralnetworks naturally within the linear unitary dynamics of a quantumcomputer. Once such duplication is achieved, one can then naturallyextend those neural networks to take advantage of the computationalproperties of quantum physics, such as superposition, entanglement, andinterference.

Most known approaches to implement QNNs struggle to reconcile nonlinearneural network transformations with the linear structure of quantumevolution that occurs in quantum computing. Several approaches may beused to address this issue. For example, one approach uses arepeat-until-success procedure, in which the nonlinear activations arecarried out by using a measurement and the process is repeated manytimes unless some desired measurement outcome is obtained. The runningtime of this repeat-until-success procedure is usually notdeterministic, i.e., it can take many repetitions to enact the desirednonlinearity. As a result, this approach has severe reliability andscalability issues, especially when implemented on quantum computershaving limited coherence time and on large fault-tolerant devices, wherethe overhead from each repeat-until-success sub-circuit can accumulate.The resulting systems may take even a longer time to carry out anonlinear function compared to classical computers. In addition, thesystem may also include many additional quantum gates or iterativeloops, thereby creating additional overhead (e.g., in terms of bothrunning time and resources).

Some approaches use photonics as the information carriers. Theseapproaches, however, are restricted to classical light withoutleveraging features and properties of quantum mechanics, such as theability to work in different bases, the ability to create superpositionand entanglement with respect to a chosen basis, quantum interferenceeffects, and the ability to manipulate and resolve individual quanta oflight (i.e., photons). Some other approaches use a photonic scheme forimplementing a quantum variant of neural networks. These approaches,however, do not propose or suggest how to use the proposed photonicarchitecture to implement a classical neural network faithfully, nor asa universal quantum computer.

Apparatus and methods described herein employ quantum photonic gates toimplement linear and nonlinear transformations with little or nooverhead. By leveraging the unique properties of quantum photonicsystems, apparatus and methods described herein can implement thenonlinear neural network transformation x→φ(Wx+b) in a reversiblemanner, and with various choices of activation function available.Furthermore, the resulting device can be very flexible, free ofarchitectural or theoretical restrictions on the width or depth of theneural network that can be implemented.

FIG. 1 shows a schematic of an apparatus 100 for universal quantumcomputing and machine learning, according to an embodiment. Theapparatus 100 includes a plurality of processing layers 110 coupled inseries (only one processing layer 110 is shown in FIG. 1 forillustration purposes). The output from one processing layer istypically sent to the next processing layer as the input (see, e.g.,FIG. 3 ). Each processing layer 110 includes a Gaussian unit 120 and anonlinear unit 130. The Gaussian unit 120 is configured to perform alinear transformation on an input signal 101 including a plurality ofoptical modes. To this end, the Gaussian unit 120 includes a network ofinterconnected beamsplitters 128 and phase shifters 126 and squeezers122 operatively coupled to the network of interconnected beamsplitters128 and phase shifters 126. The nonlinear unit 130 includes a pluralityof nonlinear gates operatively coupled to the Gaussian unit 120 andconfigured to perform a nonlinear transformation on the optical modes.The apparatus 100 also includes a controller 140 operatively coupled tothe processing layers 110 and configured to control the setting of theprocessing layers 110.

In some embodiments, the Gaussian unit 120 can also include displacers124 operatively coupled to other components in the Gaussian unit 120.The components in the Gaussian unit 120 can be arranged in various ways.In some embodiments, the Gaussian unit 120 can apply squeezing using thesqueezers 122 on all optical modes in the input signal 101, followed bypassing the input signal through an interferometer formed by the networkof interconnected beamsplitters 128 and phase shifters 126. Then thedisplacers 124 are used to apply a displacement on all optical modes inthe input signal 101. In some implementations, the beamsplitters 128 andthe phase shifters 126 can form two linear interferometers with thesqueezers 122 disposed in between. The displacers 124 are then used toapply displacement to the optical modes in the input signal 101 (see,e.g., FIG. 2 and descriptions below).

In some embodiments, the apparatus 100 also includes a light source 150to provide the input signal 101 that includes an array of optical modes.Alternatively, the input signal 101 can be provided from a communicationchannel in optical communication with the apparatus 100. In someembodiments, the input signal 101 can be provided by another photonicdevice, a physical quantum system, or an upstream processing layer,among others. In some embodiments, the input signal 101 can include anencoded optical signal. In some embodiments, the input signal 101 caninclude an electronic signal and then be converted into an opticalsignal within the apparatus 100 by a converter (not shown in FIG. 1 ).

In some embodiments, the input signal 101 can include optical signalswithout encoding (i.e., vacuum states). In these embodiments, the firstone or more processing layers can be used to prepare the desired inputstates (e.g., via squeezing and displacement). More details can be foundbelow with references to FIGS. 6 and 7 .

In some embodiments, the apparatus 100 includes an input interface toreceive the input signal 101 and couple the input signal 101 into theprocessing layers 110. For example, the input interface can include anarray of waveguides to receive the input signal 101 that includes anarray of optical modes (e.g., a spatial array). In some embodiments, theinput signal 101 can be sent into the processing layers 110 without aseparate input interface. For example, the waveguides in thebeamsplitters 128 can be used as the input interface. In someembodiments, the apparatus 100 can include a separate output interface(not shown in FIG. 1 ) to send out the output signal 102. In someembodiments, the output signal 120 can be sent out of the processinglayers 110 via waveguides that are part of the processing layers 110.

In some embodiments, at least a portion of the apparatus 100 can beconstructed based on integrated photonics. For example, the processinglayers 110 and the light source 150 can be fabricated on the samesubstrate (e.g., semiconductor, sapphire, etc.). In some embodiments,the apparatus 100 can be constructed using bulk optics.

In some embodiments, the apparatus 100 further includes a detection unit(not shown in FIG. 1 ) configured to measure the output signal 102. Insome embodiments, the detection unit includes an array of single photondetectors (SPDs), each of which is configured to measure the photonnumber in a corresponding optical mode in the output signal 102. In someembodiments, the detection unit can include a homodyne detector and/or aheterodyne detector.

In some embodiments, the nonlinear gates in the nonlinear unit 130 caninclude cubic phase gates or Kerr gates. Different nonlinear gates canbe used within the same processing layer 110 or within the apparatus100.

In some embodiments, the nonlinear gates can be replaced with nonlinearchannels or nonlinear operations that are carried out by measurement ofa subset of one or more optical modes, followed by conditionaltransformations on a subset of the modes or a post-selection on certainmeasurement events. The conditional transformation can be, e.g., anapplication of squeezers with parameters set based on processing of theoutput measurements. The post-selection can be, e.g., on the event ofacquiring single photons in each measured mode during a Fock (i.e.,photon-number) basis measurement. The conditional transformations can becarried out in several ways. In some implementations, the conditionaltransformation can be performed on the modes that are not measured. Insome embodiments, the conditional transformation can be performed onmodes that are measured. In some embodiments, the conditionaltransformation can be performed on some modes that are measured and somemodes that are not measured.

If the conditional transformations are performed on modes that aremeasured, new inputs are injected into the next layer to replace themeasured modes, and the states of these input modes can be conditionalon the measurement results from the previous layer. The resulting effectof these operations is the application of a non-Gaussian (or nonlinear)transformation on the state of the optical system via ameasurement-induced process.

The controller 140 in the apparatus 100 is configured to control thesetting of the processing layers 110. The controller 140 can beoperatively coupled to each beamsplitter 128 and phase shifter 126 inthe network of interconnected beamsplitters and phase shifters, eachsqueezer 122, each displacer 124 (if included in the apparatus 100), andeach nonlinear gate 130, thereby increasing the degree of control overthe processing layers 110. In some embodiments, the change of thesetting can be achieved via electrical signals (e.g., change the voltageapplied over phase shifters 126), and the controller can include aclassical computer or processor.

In some embodiments, the controller 140 is operatively coupled to theoutput of the apparatus 100. For example, information of the outputsignal 102 is sent to the controller 140, which in turn changes thesetting of the processing layers 110 based on the received information.This feedback scheme can be used to construct an artificial neuralnetwork, in which the controller 140 can adjust the setting of theprocessing layers 110 if the measured result of the output signal 102 isdifferent from the desired result.

In some embodiments, the apparatus 100 can be configured to implementcontinuous variable (CV) quantum computing as follows. The CV quantumcomputing leverages the wavelike properties of photons. Quantuminformation is encoded not in qubits, but in the quantum states offields, such as the electromagnetic field, thereby making it suitable tophotonic hardware. The observables in the CV picture, e.g., position{circumflex over (x)} and/or momentum {circumflex over (p)}, havecontinuous values, but qubit computations can also be embedded into thequantum field picture, so there is no loss in computational power bytaking the CV approach.

In CV quantum computing, the input signal 101 is split up into multiplespatiotemporal modes (e.g., as illustrated in FIG. 1 ). Each mode canrepresent a given path of light along an interferometer formed bybeamsplitters 128 and phase shifters 126. Multiple choices of basis areavailable for describing photonic quantum systems, including, but notlimited to: (i) the eigenstates of the quantum-mechanical positionoperator {circumflex over (x)}; (ii) the eigenstates of thequantum-mechanical momentum operator {circumflex over (p)} (i.e., thequantum Fourier transform of position); (iii) the Fock basis states{|n>}, each representing a fixed number of photons; (iv) theover-complete basis of coherent states; and (v) a dual-rail ormultiple-rail encoding where a computational basis is formed based onthe location of a single photon amongst multiple modes.

Without being bound by any particular theory or mode of operation, thetensor product of position eigenstates in an N-mode system is given by:

|x>=|x ₁ >⊗ . . . ⊗|x _(N)>,  (1)

where x=(x₁, . . . , X_(N)) is an N-dimensional real vector. The statesof an N-mode system can be written as:

|ψ>=∫dxψ(x)|x>  (2)

with the wavefunction ψ(x)ϵC^(N) satisfying ∫dx|ψ(x)|²=1.

The quantum system evolves via a unitary transformation Û_(H)=exp(−itĤ)acting on |ψ>, where Ĥ=H({circumflex over (x)}₁, . . . , {circumflexover (x)}_(N), {circumflex over (p)}₁, . . . , {circumflex over(p)}_(N)) is the Hamiltonian of the system and is a polynomial ofposition and momentum operators {circumflex over (x)}_(i) and{circumflex over (p)}_(i), respectively. Such evolutions can be realizedin the apparatus 100 through quantum photonic gates that form auniversal gate set. In some embodiments, a universal gate set includestwo-mode beamsplitters 128 and the following single mode gates:squeezers 122, displacers 124, phase gates 126, and a single fixed classof nonlinear gates 130. Furthermore, the beamsplitters 128 and phaseshifters 126 can be combined to create an N-mode linear-opticalinterferometer. Any appropriate combination of interferometers,squeezers, displacers and phase shifters can perform a so-calledGaussian transformation (and accordingly such a combination is referredto as a Gaussian unit herein).

In some embodiments, a fixed nonlinear gate 130 (also referred to as anon-Gaussian unit) can be generated by any Hamiltonian with a polynomialdegree of 3 or higher. The combination of the Gaussian unit 120 with asingle fixed nonlinear gate 130 can be used for universal CV quantumcomputation, i.e., any unitary generated by a Hamiltonian, which ispolynomial in {circumflex over (x)} and {circumflex over (p)} can beconstructed by building up from these elementary gates using apolynomial-depth circuit.

The single-mode elementary gates (e.g., squeezers 122, displacers 124,and phase shifters 126) and two-mode beamsplitters 128 have a number ofcontrollable free parameters, which can be used to alter the unitarytransformation performed by the processing layers 110. These parametersinclude, for example, squeezing factor of the squeezers 122, amount ofdisplacement applied by the displacers 124 (in the position and/ormomentum plane), phase shift introduced by the phase shifters, and splitratio of the beamsplitters 128. The configuration of these parameters isalso referred to as the setting of the processing layers 110.

In some embodiments, the setting of the processing layers 110 can bepredetermined. In some embodiments, the setting of the processing layers110 can be set dynamically, e.g., following a machine learning paradigm.

In some embodiments, a user can control these parameters to select thequantum algorithm to be implemented by the apparatus 100 (also referredto as operation mode of the apparatus 100, such as quantum computing,CNN, or QNN). For example, the user can select the operation mode viathe controller 140, which can include a classical computer with a userinterface to facilitate the mode selection.

The optical modes described above are represented by theirwavefunctions. Alternatively, an equivalent representation of opticalmodes can be given by Wigner functions, which represent the state of agiven mode as a quasi-probability distribution in the phase space ofposition and momentum operators. While unitary evolutions of a quantumsystem are generally a linear transformation on wavefunctions, thecorresponding Wigner function transformation can be nonlinear. Gaussiantransformations perform a linear (or affine) phase space transformation,while non-Gaussian transformations can perform non-linear phase spacetransformations. On the other hand, optical interferometers are oftenreferred to as linear.

It is also worth noting that the unitary transformations carried out bylinear optical interferometers (e.g., formed by beamsplitters 128 andphase shifters 126) are not the same as the unitary transformationscarried out by the full layers 110 of the apparatus 100. Morespecifically, the interferometers carry out unitary transformations onthe collection of creation and annihilation operators of the opticalmodes (or, equivalently, on the quadrature operators). Thesetransformations correspond to finite-dimensional unitary matrices. Incontrast, the full layers 110 enact unitary transformations on thewavefunctions, i.e., in the continuous space of square-integrablefunctions. These transformations correspond to infinite-dimensionallinear transformations. As used here, the nonlinear gates 130 correspondto gates enacting non-Gaussian transformations.

In some embodiments, the apparatus 100 is configured as ageneral-purpose photonic quantum computer, which is a reconfigurablecomputing device that can perform a range of user-defined processes andmethods through transformations and measurements on a quantum system.For example, the apparatus 100 can be switched between three operationmodes: classical neural network (CNN), quantum computing (QC), andquantum neural network (QNN).

To implement a CNN, the apparatus 100 can be configured as follows. Thenetwork of interconnected beamsplitters 128 and phase shifters 126 canbe divided into two portions. The first portion forms a firstinterferometer to apply a first phase-less Gaussian transformation onthe optical modes, followed by the squeezers 122 to apply aposition-only squeezing (i.e., without squeezing in the momentum) to theoptical modes. The second portion of the network of interconnectedbeamsplitters 128 and phase shifters 126 forms a second interferometerto apply a second phase-less Gaussian transformation on the opticalmodes after the squeezers 122. The displacers 124 are configured toapply a position-only displacement to the plurality of optical modes(i.e., no displacement in the momentum). In addition, the nonlineargates 130 can be configured to apply the nonlinear transformationbetween a first set of position eigenstates and a second set of positioneigenstates of the optical modes. More details of the CNN mode can befound below with reference to FIG. 6 .

To implement QC, the apparatus 100 can be configured as follows. A firstportion of the network of interconnected beamsplitters 128 and phaseshifters 126 is configured to form a first interferometer to apply afirst phase-sensitive Gaussian transformation on the optical modes inthe input signal 101, followed by the squeezers 122 configured to applysqueezing along an arbitrary axis in a position-momentum plane to theoptical modes. A second portion of the network of interconnectedbeamsplitters 128 and phase shifters 126 is configured to form a secondinterferometer to apply a second phase-less transformation on theoptical modes. The displacers 124 are configured to apply a displacementin position and momentum to the plurality of optical modes, and thenonlinear gates 130 are configured to apply an arbitrary nonlineartransformation (e.g., cubic phase gate or Kerr gate) to the plurality ofoptical modes. In some embodiments, the apparatus 100 can implement QCby creating quantum effects (e.g., entanglement or superposition) in theoptical modes. These quantum effects can be created by the Gaussian unit120 and/or the nonlinear gates 130.

To implement a QNN, the apparatus 100 can be configured as follows. TheGaussian unit 120 and the nonlinear gates 130 can be configured the sameway for QC as described above. In addition, the controller 140 isconfigured to change the setting of the processing layers 110 based onthe output signal 102 from the processing layers 110. More details aboutthe QNN mode can be found below with reference to FIG. 7 .

In some embodiments, the change of the operation mode (e.g., between QC,CNN, and QNN) can be achieved manually by a user. For example, the usercan manually change the setting of the processing layers. In someembodiments, the change of the operation mode can be achieved by thecontroller 140. Once the user selects a given mode, the controller 140can automatically adjust the setting of the processing layers 110 toswitch the apparatus 100 into the given mode.

In some embodiments, the processing layers 110 can be configured toimplement a transformation that preserves a certain subset of CV quantumstates. This subset can be discrete, e.g., as defined by a dual-rail ormultiple-rail encoding, thereby allowing for the encoding of qubit-and/or qudit-based quantum systems within the quantum optics hardware inthe apparatus 100. The configuration also allows for universal qubit- orqudit-based quantum computation by concatenating layers.

In other words, this configuration allows qubit based quantumcomputation within the CV setting. In some embodiments, one can encodequbits using the dual rail encoding. For example, the |10> state of 1photon in mode 1 and 0 photons in mode 2 can be mapped to the qubitstate |0>, while |01> can be mapped to the qubit state |1>. The set ofCV gates can then be configured to map within the |10> and |01> subspaceso as to implement qubit-based computation. For example, this can beachieved by concatenating gates and layers together. In someembodiments, the concatenation structure can be configured by usingmachine learning, i.e., one can learn how the layers enact a gate (e.g.,a Hadamard gate) on that qubit subspace.

FIG. 2 shows a schematic of an apparatus 200 including twointerferometers at each processing layer for quantum computing andmachine learning, according to an embodiment. The apparatus 200 includesprocessing layers 210 connected in series (only one processing layer 210is illustrated in FIG. 2 ). Each processing layer includes an array ofsqueezers 222 disposed between a first interferometer 226 a and a secondinterferometer 226 b. Each of the two interferometers 226 a and 226 bcan be constructed by beamsplitters and phase shifters. In someembodiments, the beamsplitters and phase shifters can form aninterferometer via the canonical Reck encoding. More details of Reckencoding can be found in, for example, M. Reck, A. Zeilinger, H. J.Bernstein, and P. Bertani, “Experimental realization of any discreteunitary operator,” Physical review letters, vol. 73, no. 1, p. 58, 1994,which is incorporated by reference in its entirety. In some embodiments,the beamsplitters and phase shifters can form an interferometer via theClements encoding. More details of Clements encoding can be found in,for example, W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S.Kolthammer, and I. A. Walmsley, “Optimal design for universal multiportinterferometers,” Optica, vol. 3, no. 12, pp. 1460-1465, 2016, which isincorporated by reference in its entirety.

The processing layer 210 also includes an array of displacers 224disposed after the second interferometer 226 b, followed by an array ofnonlinear gates 230. In some embodiments, the nonlinear gates 230 caninclude the cubic phase gate or the Kerr gate. A controller 240 isoperatively coupled to the processing layer 210 to control the settingof the processing layer 210. In some embodiments, the apparatus 200 caninclude a light source 250 to provide an input signal 201 including anarray of optical modes. In some embodiments, the controller 240 is alsooperatively coupled to output of each processing layer 210 andconfigured to adjust the setting of each processing layer 210 based onattributes of the output signal 202. The settings to operate theapparatus 200 in different modes (e.g., QC, CNN, and QNN) can besubstantially the same as for the apparatus 100.

FIG. 3 shows a schematic of an apparatus 300 including a feedback loopfor quantum computing and machine learning, according to an embodiment.The apparatus 300 includes a series of processing layers 310(1), 310(2),. . . , and 310(N) (collectively referred to as processing layers 310),where N is a positive integer. Each processing layer 310(1) through310(N) can be substantially identical to the processing layer 110 shownin FIG. 1 . An input signal 301 is processed by the processing layers310(1) to 310(N) to produce an output signal 302, which is then sentback, via a feedback loop 360, to the input side of the processinglayers 310 as anew input signal. The apparatus 300 also includes acontroller 340 operatively coupled to each processing layer 310(1) to310(N) and configured to control the operation of the feedback loop 360as well.

Operations of the apparatus 300 include reading in the problem/processdescription and data, performing processing (e.g., by the processinglayers 310), and measuring the output 302 of the processing. Theprocessing stage on the apparatus 300 involves application of anarbitrary unitary, i.e., generated by a Hamiltonian that is a polynomialof position and momentum operators to an arbitrary degree. An arbitraryunitary can be enacted by repeatedly applying the layer structure of theapparatus 300 with variable gate parameters at each layer 310(1) to310(N).

In some embodiments, the feedback loop 360 can include one or morewaveguides to send the output signal 302 (or a portion of the outputsignal 302) back to the input side of the processing layers 310. In someembodiments, the feedback loop 360 can include optical fibers,free-space optical elements (e.g., mirrors), and any other appropriatemedium. In some embodiments, the output signal 302 can be converted intoelectrical signals (e.g., electronic data) and then sent back to theinput end of the processing layers 310. A converter is then used toconvert the electronic data back to optical signal, which is then sentto the processing layers 310 for further processing.

FIG. 4 is a flowchart illustrating a method 400 of quantum computingwith a feedback loop, according to an embodiment. The method 400includes, at 410, reading in the problem to be solved (e.g., receivingdata describing the problem). At 420, initial input states are preparedfor the quantum computing (e.g., using the apparatus 100 shown in FIG.1, 200 shown in FIG. 2 , or 300 shown in FIG. 3 ) based on the receiveddata.

The steps 410 and 420 are also collectively referred to as theproblem/process description and data read-in phase. In some embodiments,this read-in phase can be achieved by controlling input light to preparethe proper input states that enter the processing layers in thecomputing devices. In some embodiments, this read-in phase can beachieved by dynamically controlling the gate parameters of quantumphotonic gates in one or more of the processing layers to encode theinformation that is read in at 410. In some embodiments, the read-inphase can be achieved by directly feeding the layer structure of thecomputing device with an output from another quantum device or a quantumcommunications network (e.g., in the form of quantum data).

The method 400 also includes, at 430, processing of the input states byperforming loops through the processing layers. In some embodiments, theprocessing stage can be performed with one or more repetitions of thelayer structure by using an electronic controller (e.g., 140 in FIG. 1,240 in FIG. 2 , or 340 in FIG. 3 ) to set gate parameters such that adesired quantum algorithm is implemented. In some embodiments, the usercan also have the option not to apply certain quantum photonic gates ina layer by, e.g., setting the corresponding parameters to zero.

The method 400 also includes, at 440, determining whether a maximalnumber of loops has already been performed. If not, the method 400returns back to step 430 for further processing through the processinglayers. If a sufficient number of loops is already performed, the method400 moves forward to 450 for output processing so as to reach problemsolution at 460.

The output from the computing device (e.g., quantum computer) can beaccessed through measurements. In some embodiments, the measurements canbe performed on one or more modes after the final processing layer. Insome embodiments, the measurements can be performed on a subset of oneor more modes after the application of an intermediate processing layer.In these embodiments, the measured data can be processed and then fedforward into the computing device as the processing continues.

Feed-forward of processed measurement information can take severalforms. In some embodiments, the values of gate parameters (e.g.,squeezing or displacement amplitudes) in a downstream layer aredetermined based on the measurement result. In some embodiments, a newquantum state is prepared and the form of the new quantum state is basedon the measurement result. The new quantum state is then injected into asubset of modes in a downstream layer. Once all processing has ceasedand measurement data has been extracted, the data may then be processedby a classical computer to yield the output of the quantum computer.

The apparatus described herein (e.g., 100-300 in FIGS. 1-3 ) can alsofunction as part of a quantum/classical hybrid system, which accordinglycan leverage both quantum and classical computing resources.Quantum-classical hybrid computing devices can be a promising approachto satisfy near-term quantum computing, because they allow quantum andclassical computers to perform tasks where they have best efficiency.For example, classical computers can perform sorting tasks efficiently,while quantum computers can perform prime factorization in a moreefficient way.

Quantum photonic hardware described herein is able to operate atwavelengths typically used in classical optical telecommunications(e.g., about 1550 nm). Therefore, this quantum photonic hardware can bereadily coupled to existing communication infrastructure usingcommercially available instruments (e.g., fibers).

FIGS. 5A-5D shows schematics of systems for quantum-classical hybridcomputing, according to embodiments. In these schematics, solid linesrepresent quantum information and dashed lines represent classicalinformation. The hashed boxes represent interfaces between quantuminformation and classical information. Quantum computers in FIGS. 5A-5Dcan include any of the apparatus described herein with references toFIGS. 1-3 above.

FIG. 5A shows a schematic of a system 501 including a classical computer511 and a quantum computer 521 connected in series. Classical signalsprocessed by the classical computer 511 are fed into the quantumcomputer 521 for further processing via one of the interfaces 531. Insome embodiments, the interfaces 531 can include a “direct plug-in”using optical fibers that connect classical and quantum photonicdevices. In some embodiments, the interfaces 531 include a device thatfurther includes measurement of quantum signals and control of quantumphotonic lasers and gates by a classical processor. In some embodiments,the interfaces 531 can include one or more processing layers describedherein (e.g., 110 in FIG. 1 ) to facilitate the signal transmissionbetween quantum and classical devices. For example, the processing layerin the interfaces 531 can receive vacuum state as the input and add ahigh degree of squeezing to each mode in the input. The processing layercan also add displacement in position to each mode. The system 501illustrated in FIG. 5A can be used when it is more efficient to performa partial processing of classical data before feeding the processed datato the quantum component, such as classification of credit card data asfraudulent or genuine.

FIG. 5B shows a schematic of a system 502 including a quantum computer522 to perform pre-processing and send the pre-processed data to aclassical computer 512 for further processing. Interfaces 532 are usedto perform the data transfer between the classical computer 512 and thequantum computer 522. The system 502 can be used, for example, when theoutput signal from the quantum computer 522 is interpreted by theclassical computer 512.

FIG. 5C shows a schematic of a system 503 including a classical computer513 and a quantum computer 523 connected in parallel. Interfaces 533 areused to transmit data between the classical computer 513 and the quantumcomputer 523. In the system 503, the classical computer 513 and thequantum computer 523 act in parallel, with the potential for two-waydata exchange during processing. The system 503 can be useful forapplications involving a continuous two-way data stream.

FIG. 5D shows a schematic of a system 504 including a dedicated quantumprocessing unit 524 with a classical controller 514. In the system 504,the classical controller 514 can use the quantum processing unit 524 fordedicated processing of certain subroutines. In some embodiments, theclassical controller 514 can assign multiple computations to the quantumprocessing unit 524 before a classical signal is read-out.

The system 504 can be used when the quantum processing unit 524 is usedto calculate an intractable quantity, such as the calculation ofvibronic spectra in chemistry. In these applications, the problemstatement can be declared by classical computers, but the actualcalculations are usually challenging or impractical for classicalcomputers to perform.

As described herein, the computing apparatus shown in FIGS. 1-3 can alsobe used to implement both classical and quantum neural networks, withapplications in machine learning and optimization. In general, operationas a classical neural network (CNN) can be achieved by placing certainrestrictions on the parameter range of some quantum photonic gates inthe layer structure, as well as inputting light and measuring in a fixedbasis. Relaxing these constraints allows operation of the apparatus as ageneral QNN, which can be trained efficiently and also controlled toemulate certain common architectures of CNN layers, such as theconvolutional layer.

FIG. 6 shows a schematic of a classical neural network (CNN) 600implemented by apparatus 100-300 shown in FIGS. 1-3 , according to anembodiment. The CNN 600 includes multiple processing layers 610(1) to610(5) (collectively referred to as processing layers 610) to form aneural network and process an input signal 601. The processing layers610 can be divided into three groups: an input layer 620 formed by thefirst processing layer 610(1), multiple hidden layers 630 formed byprocessing layers 610(2) through 610(4), and an output layer 640 formedby the last processing layer 610(5). Five processing layers 610 areshown in FIG. 6 for illustrating purposes only. In practice, any othernumber of processing layers 610 can also be used.

In some embodiments, each processing layer 610(1) to 610(5) can besubstantially identical to any of the processing layers described herein(e.g., 110 in FIG. 1, 210 in FIG. 2 , and 310 in FIG. 3 ). In someembodiments, each of the processing layers 610(2) through 610(4) can besubstantially identical to any of the processing layers describedherein. The input layer 620 and the output layer 640 can use other typesof layers (e.g., without squeezers and/or displacers).

In the CNN 600, each layer 610(1) through 610(5) corresponds directly toa neural network layer and each mode of light in the input signal 610corresponds to a neuron. Operation of the processing layers 610 as a CNNincludes loading in some N-dimensional classical data xϵR^(N). This canbe achieved by inputting the N-mode tensor product of position states|x> according to Equation (1). The state |x> can be approximatelyprepared by input laser light and/or using an input layer structure withhigh position squeezing and a position displacement of x_(i) along modei.

A fully connected feed-forward CNN layer (e.g., 610(2) to 610(4)) mapsan N-dimensional vector to an M-dimensional vector by performing thetransformation L(x)=φ(Wx+b), where WϵR^(M×N) (representing atransformation matrix having the dimension of M×N), bϵR^(M) is a biasvector having M elements, and co is an elementwise non-linear function(such as rectified linear units or the sigmoid function). The layers 610can perform the transformation |x>→|φ(Wx+b)> in the following way.

First, W can be decomposed using the singular value decomposition asW=O₂ΣO₁, where Σ is a diagonal matrix with positive entries and O_(i)(i=1, 2) are orthogonal matrices. Each layer in the processing layers610 first applies O₁ by restricting the first set of beamsplitters andphase gates (forming an interferometer) to phase-less transformations,resulting in |x>→>|O₁x>. Next, squeezers in the processing layer areapplied with a restriction to position-only squeezing, allowing for thetransformation |O₁x>→|ΣO₁x> by appropriately controlling gate parametersof the squeezers. The second set of beamsplitters and phase gates(forming another interferometer) is also restricted to phase-lesstransformations, resulting in |ΣO₁x>→|O₂ΣO₁x>=|Wx>. When the input andoutput dimensions of W do not match (i.e., M is not equal to N), thelayer structure is composed of max{M, N} modes.

Addition of a bias vector b can be achieved by restricting displacers oneach mode to position-only displacement. This results in thetransformation |Wx>→|Wx+b>. Finally, a nonlinear function φ is appliedthrough nonlinear gates or channels acting on each mode in the inputsignal 601, with the restriction that the nonlinear transformations mapposition eigenstates to position eigenstates, i.e., |x_(i)>→|φ(x_(i))>.This restriction can ensure that the processing through the layers 610does not generate quantum effects, such as superposition orentanglement, amongst the position eigenstates, thereby maintaining theclassical nature of the CNN 600. The result of the nonlinear gates isthe transformation |Wx+b>→|φ(Wx+b)>. Compounding each of the operationsabove, each of the layers 610 is able to perform the CNN layertransformation on position eigenstates:

|x>→|L(x)>  (3)

The output of a given layer 610(1) through 610(5) can be read out asL(x) by performing a homodyne position measurement on each of the modes.By fixing the constrained parameters described above, a user can apply apredetermined function to the input signal 601. Instead, by allowing theconstrained parameters to be variable, the user can operate the layers610 as a CNN for machine learning. Training of the layer structure as aCNN can be performed using standard numerical or automaticdifferentiation techniques in combination with a classical device. Forexample, the training can be carried out by performing one forward passthrough the layers 610, computing gradients using a numerical estimationtechnique (e.g., the finite-difference method) or the backpropagationalgorithm (leveraging the classical coprocessor as needed), and thenusing the output to update parameters in the layers 610.

FIG. 7 shows a schematic of a quantum neural network (QNN) 700implemented by apparatus shown in FIGS. 1-3 , according to anembodiment. The QNN 700 includes multiple processing layers 710(1) to710(5). The first processing layer 710(1) is configured as an inputlayer 720, the last processing layer 710(5) is configured as an outputlayer 740, and the middle layers 710(2) to 710(4) are configured ashidden layers 730. Each of the processing layers 710 can besubstantially identical to any of the processing layers described herein(e.g., 110 in FIG. 1, 210 in FIG. 2, and 310 in FIG. 3 ).

The QNN 700 can operate with the following setting. Each processinglayer 710 includes a first interferometer (e.g. formed by interconnectedbeamsplitters and phase shifters) configured to apply a firstphase-sensitive Gaussian transformation on the optical modes, followedby squeezing along an arbitrary axis in the position-momentum plane tothe optical modes (e.g., using squeezers). Then a second interferometer(e.g., formed by another set of interconnected beamsplitters and phaseshifters) is used to apply a second phase-less transformation on theoptical modes. In addition, a displacement in position and momentum tooptical modes can also be applied to the optical modes (e.g., usingdisplacers). The nonlinear gates in each processing layer are configuredto apply an arbitrary nonlinear transformation to the optical modes.

Moreover, the input data and output data can also be configured in moregeneral forms. More specifically, the user of the QNN 700 can choose anybasis, including but not limited to, the momentum eigenbasis, the Fockbasis, the overcomplete basis of coherent states, or thedual-/multiple-rail basis. At input, different bases can be controlledby adjusting the input lasers and/or using one or more input layerstructures with arbitrary gate parameters. At output, different basescan be controlled by passing through output layer structures and/orperforming measurements in different bases (e.g., the Fock basis).Quantum photonics used in the QNN 700 allows for homodyne, heterodyne,and photon-number counting measurements. In the QNN 700, thecorrespondence between artificial neurons and optical modes isgeneralized to the quantum realm: several different representations areavailable for encoding and interpreting data, e.g., using Fock basisstates, or a basis of nonorthogonal coherent states.

The QNN 700 has a number of advantages. First, the structure of gatesused in the QNN 700 can be substantially similar to gates used in a CNN(e.g., the CNN 600) but the operation of the QNN 700 allows for greatergenerality in the input/output data and the parameters of gates. As aresult, the QNN 700 can take advantage of the quantum properties ofsuperposition and entanglement. Second, as described with reference toFIGS. 1-3 , each processing layer 710 includes a universal set ofquantum photonic gates and can perform universal quantum computing,which is challenging to simulate on a classical device. The QNN 700 canbe used to perform a wide range of quantum algorithms, including thosethat are known or believed to be intractable on classical devices.Third, the QNN 700 can perform reversible nonlinear transformations onprobability distributions by making use of quantum interference effects.For comparison, CNNs are typically used to perform reversible lineartransformations on probability distributions. Reversible nonlineartransformations can be useful in machine learning because they allow forgreater power in expressing or representing various probabilitydistributions.

Training of the QNN 700 can be achieved by an iterative procedure. Thisprocedure includes, for example, defining a cost function and theniteratively updating the gate parameters in the processing layers 710until the cost function is minimized. The minimization can be achievedvia at least two approaches. First, the minimization can be achievedthrough simulation of the quantum hardware on classical computingdevices, which allow for standard automatic differentiation techniquesusing the functional representation of the QNN. This option can beachieved on small-scale systems (i.e., a small mode number and/or layernumber) but can quickly become impractical due to the overhead ofquantum simulation as the system scale increases.

Second, the minimization can be achieved by direct evaluation ofgradients on quantum hardware. This option allows for evaluation of thegradient of the cost function with respect to each gate parameter in thelayer structures.

FIG. 8 is a flowchart illustrating a method 800 of evaluating gradientswith respect to a QNN, according to an embodiment. Direct evaluation ofgradients can be achieved through both analytic and numeric evaluationof derivatives. The method 800 allows a user or a controller to choosewhich approach to use. In the method 800, an observable f(θ) is definedas the output of the QNN at 810. The observable f(θ) is a function of agate parameter θ and is associated with the cost function.

The method 800 also includes performing gradient descent to find theoptimal θ to optimize the cost function. This includes evaluation of∂_(θ)f(θ) at 820. For many types of gates, including Gaussian gates, anexact expression for the derivative can be theoretically derived, and∂_(θ)f/(θ) can be evaluated using an analytic derivative formula.Without being bound by any particular theory or mode of operation, thisanalytic expression has the form

∂_(θ) f(θ)=c[f(θ+s)−f(θ−s)]  (4)

where with cϵR and sϵR are fixed parameters dependent upon the gates.

Equation (4) can be evaluated using the same QNN hardware by simplyrunning the circuit with the gate parameter shifted by ±s. Due to thenature of quantum systems, evaluating f(θ±s) (as an expectation value)involves multiple runs of the QNN to arrive at a good estimation.

On the other hand, gradients ∂_(θ)f(θ) can be estimated numericallyusing the finite difference method:

$\begin{matrix}{{\partial_{\theta}{f(\theta)}}\frac{{f\left( {\theta + {\frac{1}{2}\Delta\mu}} \right)} - {f\left( {\theta - {\frac{1}{2}\Delta\mu}} \right)}}{\Delta\mu}} & (5)\end{matrix}$

for a small Δμ. The approach of evaluating Equation (5) can be similarto the analytic derivative method in that both approaches involve runsof the QNN with parameters shifted by ±Δμ/2. The difference between thetwo approaches is that the analytical approach is an exact gradientformula, using shift and scale parameters c and s that are determined bythe gate where the parameter θ appears. In contrast, the numericalapproach is an approximate gradient formula, using a smalluser-specified value Δμ. When using the numerical estimation technique,there is an additional error in the approximation of ∂_(θ)f(θ).

Referring back to FIG. 8 , the method 800 also includes, at 830,determining whether the gradient expression is known. If the expressionof the gradient is known, the method 800 then proceeds to 840, whereanalytical approach is used to evaluate ∂_(θ)f(θ). If, however, thegradient expression is not known, then the analytical approach may notbe available. In this case, the method 800 proceeds to 850, where thenumerical approach is used to evaluate ∂_(θ)f(θ) via Equation (5).

FIG. 9 shows a convolutional processing layer 900 that can be used forquantum computing and machine learning, according to an embodiment. Inthe convolutional layer 900, neurons are restricted to have only localconnections and the connections are in general spatially homogeneous. Asillustrated in FIG. 9 , each neuron is represented by an optical mode901(1) to 901(6) in the input signal 901, and each mode has connectionswith two output modes (for 901(1) and 901(6) at the edge) or threeoutput modes (for 910(2) to 910(5). Reversely, each output mode 902(1)to 920(6) also has only connections with two or three input modes.

The layer 900 is also referred to as a “kernel” that is set to sweepacross the data from the previous layer (e.g., image data). This can bereplicated using the structure of the layer 210 shown in FIG. 2 byconstraining the Gaussian elements (e.g., squeezers, displacers,beamsplitters, phase gates) to be generated by an overall Hamiltonianthat is translationally invariant. In other words, if H{circumflex over( )}=H(x{circumflex over ( )}, p{circumflex over ( )}), thenH{circumflex over ( )} does not change if (x{circumflex over ( )}_(i),p{circumflex over ( )}_(i)) of mode i is mapped to (x{circumflex over( )}_(i+1), p{circumflex over ( )}_(i+1)) of mode i+1.

FIG. 10 shows a recurrent processing layer 1000 that can be used forquantum computing and machine learning, according to an embodiment.Recurrent layers are often used for problems involving sequences ofdata, such as natural language or time series. The recurrent layer 1000shown in FIG. 10 take two types of input at each layer i. The first typeof data is an external input x^((i)) 1001 a, which may be from an inputdata source. The second type of data is an internal state h^((t)) 1001b, which comes from within the recurrent neural network but from aprevious layer.

In some embodiments, a subset of input modes (e.g., 1001 a) to the layer1000 are prepared according to a given signal. This preparation can be aquantum state, which may be prepared by using another layer structure,by controlling input laser light, or by feeding in from another quantumdevice. A subset of the remaining input modes (e.g., 1001 b) can betaken from the output of a previous layer. At the output of therecurrent layer 1000, a subset of the modes (e.g., 1002 a) is fed-outfrom the structure as an output signal. In some embodiments, the outputsignal 1002 a can be measured in a given basis (possibly by firstfeeding through another layer structure of this invention). In someembodiments, the output signal 1002 a is sent to another quantum device(e.g., as input signal). A subset of the remaining input modes (e.g.,1002 b) can then be sent to a later layer of the neural network.

FIG. 11 shows a residual processing layer 1100 that can be used forquantum computing and machine learning, according to an embodiment.Residual layers are often used in machine learning tasks and can provideshortcut connections that allow the input signal 1101 to be split intotwo parts 1101 a and 1101 b. The first part 1101 a is sent through thelayer 1110 and the second part 1101 b bypasses the layer 1110. The twoparts 1101 a and 1101 b are then recombined before being passed to thenext layer. In other words, the output signal 1102 includes portions ofthe input signal 1101.

If the layer 1110 itself performs the transformation x→L(x), then theresidual layer 1000 performs the transformation x→x+L(x). The layerstructure in the layer 1100 can carry out a residual QNN layer byapplying the unitary V=exp(ix{circumflex over ( )}_(i)⊗p{circumflex over( )}_(i+N)) to each mode in the input signal 1101 and a correspondingancilla mode (hence, involving 2N modes overall). This unitary carriesout:

V|x _(i)>⊗|0>=|x _(i) >⊗|x _(i)>  (6)

on modes i and i+N, with |0> being the zero-position eigenstate. Mode ican then sent through the QNN layer and mode i+1 is routed around theQNN layer.

In some embodiments, the residual layer 1100 can have restrictedparameters such that it simulates a general CNN layer. This results inthe 2N mode state as |L(x)>⊗|x>. Finally, a controlled-X (or SUM) gatecan be used mode-wise on modes i and i+N to create the state|L(x)+x>⊗|x>. The controlled-X gate can be carried out using Gaussianquantum gates. The output of the first N modes is hence a residual layerwith respect to information encoded in the product position eigenbasis.The second set of N modes can be preserved until the calculation hasfinished. The residual layer 1100 can also accept information in anyother basis and can also be operated with unrestricted gate parameters,thereby allowing the operation to go beyond the standard residual CNNlayer transformations by leveraging the quantum properties ofsuperposition, entanglement, and interference.

FIG. 12 is a flowchart illustrating a method 1200 of quantum computingand machine learning. The method 1200 includes propagating an inputsignal through a plurality of processing layers connected in series, andthe input signal includes a plurality of optical modes. The propagationof the input signal includes, at 1210, performing a lineartransformation on the plurality of optical modes using a Gaussian unit.The Gaussian unit includes a network of interconnected beamsplitters andphase shifters and a plurality of squeezers operatively coupled to thenetwork of interconnected beamsplitters and phase shifters. Thepropagation of the input signal also includes, at 1220, performing anonlinear transformation on the plurality of optical modes using aplurality of nonlinear gates. The output from step 1220 is then sent toanother processing layer, repeating the steps 1210 and 1220 (but atanother processing layer). The method 1200 also includes, at 1230,sending an output signal from the plurality of processing layers (e.g.,when a sufficient number of processing layers is used).

In some embodiments, performing the linear transformation at 1210includes several steps. First, a first phase-less transformation isapplied on the plurality of optical modes using a first portion of thenetwork of interconnected beamsplitters and phase shifters that isconfigured to form a first interferometer. Then, a position-onlysqueezing is applied to the plurality of optical modes using theplurality of squeezers, followed by applying a second phase-lesstransformation on the plurality of optical modes using a second portionof the network of interconnected beamsplitters and phase shifters thatis configured to form a second interferometer. In addition, the lineartransformation can also include applying a position-only displacement tothe plurality of optical modes using the plurality of displacers. Inthese embodiments, applying the nonlinear transformation at 1220includes applying the nonlinear transformation between a first set ofposition eigenstates and a second set of position eigenstates of theplurality of optical modes. In these embodiments, the method 1200 isconfigured to apply a classical neural network (CNN).

In some embodiments, performing the linear transformation at 1210includes: (i) applying a first phase-sensitive Gaussian transformationon the plurality of optical modes using a first portion of the networkof interconnected beamsplitters and phase shifters that is configured toform a first interferometer; (ii) applying squeezing along an arbitraryaxis in the position-momentum plane to the plurality of optical modesusing the plurality of squeezers; (iii) applying a secondphase-sensitive Gaussian transformation on the plurality of opticalmodes using a second portion of the network of interconnectedbeamsplitters and phase shifters that is configured to form a secondinterferometer; and (iv) applying a displacement in position andmomentum to the plurality of optical modes using a plurality ofdisplacers. In addition, the nonlinear transformation at 1220 includesapplying an arbitrary nonlinear transformation to the plurality ofoptical modes using the plurality of nonlinear gates. In theseembodiments, the method 1200 can be configured to implement quantumcomputing (QC).

In some embodiments, at least one of the linear transformation or thenonlinear transformation on the plurality of optical modes includescreating an entanglement or a superposition in the plurality of opticalmodes. This also allows the method 1200 to implement QC.

In some embodiments, in addition to the setting for implementing QC asdescribed above, the method 1200 also includes changing the phasesetting of the processing layers based on the output signal from theplurality of processing layers so as to implement a quantum neuralnetwork (QNN). For example, a detection unit can be used to measure theoutput signal and send the measured information to a controller, whichin turn changes the settings of the processing layers.

While various embodiments have been described and illustrated herein, avariety of other means and/or structures for performing the functionand/or obtaining the results and/or one or more of the advantagesdescribed herein, and each of such variations and/or modifications arepossible. More generally, all parameters, dimensions, materials, andconfigurations described herein are meant to be examples and that theactual parameters, dimensions, materials, and/or configurations willdepend upon the specific application or applications for which thedisclosure is used. It is to be understood that the foregoingembodiments are presented by way of example only and that otherembodiments may be practiced otherwise than as specifically describedand claimed. Embodiments of the present disclosure are directed to eachindividual feature, system, article, material, kit, and/or methoddescribed herein. In addition, any combination of two or more suchfeatures, systems, articles, materials, kits, and/or methods, if suchfeatures, systems, articles, materials, kits, and/or methods are notmutually inconsistent, is included within the inventive scope of thepresent disclosure.

Also, various concepts may be embodied as one or more methods, of whichan example has been provided. The acts performed as part of the methodmay be ordered in any suitable way. Accordingly, embodiments may beconstructed in which acts are performed in an order different thanillustrated, which may include performing some acts simultaneously, eventhough shown as sequential acts in illustrative embodiments.

All definitions, as defined and used herein, should be understood tocontrol over dictionary definitions, definitions in documentsincorporated by reference, and/or ordinary meanings of the definedterms.

The indefinite articles “a” and “an,” as used herein in thespecification and in the claims, unless clearly indicated to thecontrary, should be understood to mean “at least one.”

The phrase “and/or,” as used herein in the specification and in theclaims, should be understood to mean “either or both” of the elements soconjoined, i.e., elements that are conjunctively present in some casesand disjunctively present in other cases. Multiple elements listed with“and/or” should be construed in the same fashion, i.e., “one or more” ofthe elements so conjoined. Other elements may optionally be presentother than the elements specifically identified by the “and/or” clause,whether related or unrelated to those elements specifically identified.Thus, as a non-limiting example, a reference to “A and/or B”, when usedin conjunction with open-ended language such as “comprising” can refer,in one embodiment, to A only (optionally including elements other thanB); in another embodiment, to B only (optionally including elementsother than A); in yet another embodiment, to both A and B (optionallyincluding other elements); etc.

As used herein in the specification and in the claims, “or” should beunderstood to have the same meaning as “and/or” as defined above. Forexample, when separating items in a list, “or” or “and/or” shall beinterpreted as being inclusive, i.e., the inclusion of at least one, butalso including more than one, of a number or list of elements, and,optionally, additional unlisted items. Only terms clearly indicated tothe contrary, such as “only one of” or “exactly one of,” or, when usedin the claims, “consisting of,” will refer to the inclusion of exactlyone element of a number or list of elements. In general, the term “or”as used herein shall only be interpreted as indicating exclusivealternatives (i.e. “one or the other but not both”) when preceded byterms of exclusivity, such as “either,” “one of,” “only one of,” or“exactly one of” “Consisting essentially of,” when used in the claims,shall have its ordinary meaning as used in the field of patent law.

As used herein in the specification and in the claims, the phrase “atleast one,” in reference to a list of one or more elements, should beunderstood to mean at least one element selected from any one or more ofthe elements in the list of elements, but not necessarily including atleast one of each and every element specifically listed within the listof elements and not excluding any combinations of elements in the listof elements. This definition also allows that elements may optionally bepresent other than the elements specifically identified within the listof elements to which the phrase “at least one” refers, whether relatedor unrelated to those elements specifically identified. Thus, as anon-limiting example, “at least one of A and B” (or, equivalently, “atleast one of A or B,” or, equivalently “at least one of A and/or B”) canrefer, in one embodiment, to at least one, optionally including morethan one, A, with no B present (and optionally including elements otherthan B); in another embodiment, to at least one, optionally includingmore than one, B, with no A present (and optionally including elementsother than A); in yet another embodiment, to at least one, optionallyincluding more than one, A, and at least one, optionally including morethan one, B (and optionally including other elements); etc.

In the claims, as well as in the specification above, all transitionalphrases such as “comprising,” “including,” “carrying,” “having,”“containing,” “involving,” “holding,” “composed of,” and the like are tobe understood to be open-ended, i.e., to mean including but not limitedto. Only the transitional phrases “consisting of” and “consistingessentially of” shall be closed or semi-closed transitional phrases,respectively, as set forth in the United States Patent Office Manual ofPatent Examining Procedures, Section 2111.03.

What is claimed is:
 1. An apparatus, comprising: a plurality ofprocessing layers coupled in series, each processing layer in theplurality of processing layers including: a Gaussian unit configured toperform a linear transformation on an input signal including a pluralityof optical modes, the Gaussian unit including a network ofinterconnected beamsplitters and phase shifters; a plurality ofnonlinear gates operatively coupled to the Gaussian unit and configuredto perform a nonlinear transformation on the plurality of optical modes;and a controller operatively coupled to the plurality of processinglayers and configured to control a setting of the plurality ofprocessing layers.
 2. The apparatus of claim 1, wherein the Gaussianunit further includes: a plurality of squeezers operatively coupled tothe network of interconnected beamsplitters and phase shifters, eachsqueezer from the plurality of squeezers configured to apply one of aposition-only squeezing or a squeezing along an arbitrary axis in aposition-momentum plane to one optical mode from the plurality ofoptical modes.
 3. The apparatus of claim 1, wherein the Gaussian unitfurther includes: a plurality of displacers operatively coupled to aplurality of squeezers and the network of interconnected beamsplittersand phase shifters, each displacer from the plurality of displacersconfigured to apply one of a position-only displacement or adisplacement along an arbitrary axis in a position-momentum plane to oneoptical mode from the plurality of optical modes.
 4. The apparatus ofclaim 2, wherein: the Gaussian unit further includes a plurality ofdisplacers operatively coupled to the plurality of squeezers and thenetwork of interconnected beamsplitters and phase shifters, eachdisplacer from the plurality of displacers configured to apply aposition-only displacement to one optical mode from the plurality ofoptical modes, a first portion of the network of interconnectedbeamsplitters and phase shifters being configured to form a firstinterferometer to apply a first phase-less Gaussian transformation onthe plurality of optical modes, each squeezer from the plurality ofsqueezers being configured to apply a position-only squeezing to oneoptical mode from the plurality of optical modes, a second portion ofthe network of interconnected beamsplitters and phase shifters beingconfigured to form a second interferometer to apply a second phase-lessGaussian transformation on the plurality of optical modes, and theplurality of nonlinear gates being configured to apply the nonlineartransformation between a first set of position eigenstates and a secondset of position eigenstates of the plurality of optical modes.
 5. Theapparatus of claim 2, wherein: the Gaussian unit further includes aplurality of displacers operatively coupled to the plurality ofsqueezers and the network of interconnected beamsplitters and phaseshifters, each displacer from the plurality of displacers configured toapply a displacement in position and momentum to one optical mode fromthe plurality of optical modes, a first portion of the network ofinterconnected beamsplitters and phase shifters being configured to forma first interferometer to apply a first phase-sensitive Gaussiantransformation on the plurality of optical modes, each squeezer from theplurality of squeezers being configured to apply squeezing along anarbitrary axis in a position-momentum plane to one optical mode from theplurality of optical modes, a second portion of the network ofinterconnected beamsplitters and phase shifters being configured to forma second interferometer to apply a second phase-less transformation onthe plurality of optical modes, and the plurality of nonlinear gatesbeing configured to apply an arbitrary nonlinear transformation to theplurality of optical modes.
 6. The apparatus of claim 2, wherein: theGaussian unit further includes a plurality of displacers operativelycoupled to the plurality of squeezers and the network of interconnectedbeamsplitters and phase shifters, each displacer from the plurality ofdisplacers configured to apply a displacement in position and momentumto one optical mode from the plurality of optical modes, a first portionof the network of interconnected beamsplitters and phase shifters beingconfigured to form a first interferometer to apply a firstphase-sensitive Gaussian transformation on the plurality of opticalmodes, each squeezer from the plurality of squeezers being configured toapply squeezing along an arbitrary axis in a position-momentum plane toone optical mode from the plurality of optical modes, a second portionof the network of interconnected beamsplitters and phase shifters beingconfigured to form a second interferometer to apply a second phase-lesstransformation on the plurality of optical modes, the plurality ofnonlinear gates being configured to apply an arbitrary nonlineartransformation to the plurality of optical modes, and the controllerbeing configured to change the setting of the plurality of processinglayers based on an output signal from the plurality of processinglayers.
 7. The apparatus of claim 1, wherein at least one of theGaussian unit or the plurality of nonlinear gates is configured tocreate an entanglement or a superposition from the plurality of opticalmodes, and the controller is configured to change the setting of theplurality of processing layers based on an output signal from theplurality of processing layers.
 8. The apparatus of claim 1, furthercomprising: a light source configured to send the plurality of opticalmodes, the light source and the plurality of processing layers beingfabricated in a substrate.
 9. The apparatus of claim 1, furthercomprising: a feedback loop operatively coupled to an output interfaceof the plurality of processing layers, the feedback loop configured tosend at least a portion of an output signal back from the plurality ofprocessing layers to an input interface as a new input signal for theplurality of processing layers.
 10. The apparatus of claim 9, furthercomprising a detector operatively coupled to the plurality of processinglayers and configured to measure a photon number of each optical modefrom the plurality of optical modes in the output signal.
 11. Theapparatus of claim 2, wherein the controller is operatively coupled to:(i) each beamsplitter and phase shifter in the network of interconnectedbeamsplitters and phase shifters; (ii) each squeezer from the pluralityof squeezers; and (iii) each nonlinear gate from the plurality ofnonlinear gates.
 12. A method, comprising: propagating an input signalthrough a plurality of processing layers connected in series, the inputsignal including a plurality of optical modes, the propagating of theinput signal through the plurality of processing layers including:performing a linear transformation on the plurality of optical modesusing a Gaussian unit that includes a network of interconnectedbeamsplitters and phase shifters; and performing a nonlineartransformation on the plurality of optical modes using a plurality ofnonlinear gates; and sending an output signal from the plurality ofprocessing layers.
 13. The method of claim 12, wherein: the performingthe linear transformation on the plurality of optical modes includes:applying a first phase-less transformation on the plurality of opticalmodes using a first portion of the network of interconnectedbeamsplitters and phase shifters that is configured to form a firstinterferometer; applying a position-only squeezing to one or moreoptical modes from the plurality of optical modes using a plurality ofsqueezers; applying a second phase-less transformation on the pluralityof optical modes using a second portion of the network of interconnectedbeamsplitters and phase shifters that is configured to form a secondinterferometer; and applying a position-only displacement to one or moreoptical modes from the plurality of optical modes using a plurality ofdisplacers; and the performing the nonlinear transformation on theplurality of optical modes includes applying the nonlineartransformation between a first set of position eigenstates and a secondset of position eigenstates of the plurality of optical modes.
 14. Themethod of claim 12, wherein: the performing the linear transformation onthe plurality of optical modes includes: applying a firstphase-sensitive Gaussian transformation on the plurality of opticalmodes using a first portion of the network of interconnectedbeamsplitters and phase shifters that is configured to form a firstinterferometer; applying squeezing along an arbitrary axis in aposition-momentum plane to one or more optical modes from the pluralityof optical modes using a plurality of squeezers; applying a secondphase-sensitive Gaussian transformation on the plurality of opticalmodes using a second portion of the network of interconnectedbeamsplitters and phase shifters that is configured to form a secondinterferometer; and applying a displacement in position and momentum toone or more optical modes from the plurality of optical modes using aplurality of displacers; and the performing the nonlinear transformationon the plurality of optical modes includes applying an arbitrarynonlinear transformation to the plurality of optical modes using theplurality of nonlinear gates.
 15. The method of claim 12, wherein: theperforming the linear transformation on the plurality of optical modesincludes: applying a first phase-sensitive Gaussian transformation onthe plurality of optical modes using a first portion of the network ofinterconnected beamsplitters and phase shifters that is configured toform a first interferometer; applying squeezing along an arbitrary axisin a position-momentum plane to one or more optical modes from theplurality of optical modes using a plurality of squeezers; applying asecond phase-sensitive Gaussian transformation on the plurality ofoptical modes using a second portion of the network of interconnectedbeamsplitters and phase shifters that is configured to form a secondinterferometer; and applying a displacement in position and momentum toone or more optical modes from the plurality of optical modes using aplurality of displacers; the performing the nonlinear transformation onthe plurality of optical modes includes applying an arbitrary nonlineartransformation to the plurality of optical modes using the plurality ofnonlinear gates, and the method further includes changing a phasesetting of the plurality of processing layers based on the output signalfrom the plurality of processing layers.
 16. The method of claim 12,wherein: at least one of the linear transformation or the nonlineartransformation on the plurality of optical modes includes creating anentanglement or a superposition from the plurality of optical modes, andthe method further includes changing a phase setting of the plurality ofprocessing layers based on the output signal from the plurality ofprocessing layers.
 17. The method of claim 12, further comprising:sending the input signal to the plurality of processing layers using alight source, the light source and the plurality of processing layersbeing fabricated in a substrate.
 18. The method of claim 12, furthercomprising: sending at least a portion of the output signal from anoutput interface of the plurality of processing layers back to an inputinterface of the plurality of processing layers as a new input signalfor the plurality of processing layers.
 19. The method of claim 12,further comprising: measuring a photon number of each optical mode fromthe plurality of optical modes in the output signal.
 20. The method ofclaim 12, further comprising: changing a setting of the plurality ofprocessing layers using a controller, the controller being operativelycoupled to: (i) each beamsplitter and phase shifter in the network ofinterconnected beamsplitters and phase shifters; (ii) each squeezer in aplurality of squeezers; and (iii) each nonlinear gate from the pluralityof nonlinear gates.